In being a dog trainer, it is expected you will be asked a variety of questions regarding what you can and cannot teach a dog. The only correct answer boils down to articles regarding instinct; behaviors I can train into your dog, and those I can't.
As a trainer of six years who has yet to shadow a professional, I do not mind if my advice is taken with a grain of salt, but it is recommended that when I do give instructions, they be followed. I am also not bothered by clients asking "Why?" or giving me their two cents. While I teach, I am also taught. You can never learn too much, and you can never stop learning as long as you take the time to observe. This approach is one of the many qualities that makes up a good trainer, regardless of the species. I taught myself, through working with all kinds of dogs and all kinds of people, reading what I could, and using various strategies. I still keep my eyes open and absorb what is going on around me like a sponge. To this day, much of my training revolves around positivity and treats, as it seems to do the trick for the dogs who need basic obedience training. I consider myself a master in obedience and an intermediate in other aspects. I am open to tackling any canine issue with an open mind and a positive outlook.
However, there is no "one size fits all" method for training; rather, I see most dogs as individuals, with varying solutions to their problems. With this in mind, I know different methods work for different dogs. I know that what I use to teach a dog basic obedience may not work on a fearful dog I am rehabilitating, and that approach may not work for a dog-aggressive client I'm working with. This approach is key in becoming relevant as a successful individual in this field of work.
There is virtually nothing you cannot teach a dog to do, given you have employed the appropriate techniques, are calm and consistent, and have tapped into what motivates your dog the most, whether it be treats, praise, or giving the dog their favorite toy. You can absolutely teach an old dog new tricks. In one instance, I have worked with a dog and taught her around 40 tricks with both verbal and hand commands in a span of four months, after getting her when she was a month and a half old. While I don't advise bringing a dog home that young (barring you being a foster home), the earlier you start with training, the better. Your dogs mind is like clay, ready to be molded, and it is imperative to have a solid upbringing and provide proper socialization if you desire a well-behaved companion in the future.
However, your dogs instincts may stifle some of the behavior you're hoping for. For example, when looking for the perfect Schutzhund competitor, a handler will go in to see the litter, usually around the time the pups eyes and ears are open and their personalities are beginning to express themselves through how they interact with their littermates, and examine each pup for quality traits. In this sport, handlers look for pups that are "gamey" - that is, they have a high prey drive and are very motivated to play and are sensitive to stimuli. The more docile and quiet pups would be reserved for a family type environment, but probably not one that isn't very active. If the parents were reactive and gamey, it is safe bet that their offspring will inherit this trait, and even the ones that don't seem to display these characteristics would be likely to have a much higher energy level than your average dog. It is possible to encourage gamey behavior, but if not present from puppy-hood, it is unlikely that your dog will exhibit an escalating prey drive as they get older.
Another example: therapy dogs. Dogs who are bred specifically for the program (seeing eye dogs, general therapy dogs, dogs that assist those in wheelchairs or that are otherwise impaired) will produce litters that include pups that either make the cut, or just don't. Therapy dogs are inherently gentle, intelligent, and calm. Some pups will be more energetic, mouthy, and rough, and these are the pups that are usually placed in pet homes. Handlers look for dogs with a quiet demeanor who aren't easily distracted. In their line of work, these dogs have to be unflappable. Before they are placed in homes for their training, they are put through a rigorous series of tests documenting their personalities.
This is a good video documenting what general therapy dogs will go through prior to earning their certification, whether as an adult or a pup. They'll be faced with many scary things - opening umbrellas, loud strangers, cans with coins or rocks in them being shaken, and anything unnerving in general. The ones that pass are the ones who are instinctively calm, observant, and politely curious. They cannot be easily alarmed or show aggression. Imagine what kind of a lawsuit someone would have on their hands if they placed a dog with those traits - a service dog fleeing from a car and dragging someone in a wheelchair down the road or knocking over their blind handler and bolting, or biting a senior who gets a little rough during a hospital visit.
I am going to cite a few frequently asked questions and regarding training issues on naturally ingrained behaviors, my experience with the subject, and what I usually give in response. I have broken this post down into one issue at a time to make it easier to read. The following posts are advice I would personally give to a client in the same situations.
If you feel you have anything to add to what I have to say, go ahead and post, and I will edit each post and cite your advice. If you have no experience in the area, please refrain from commenting. There will probably be about four separate sections in this specific post, and I hope you enjoy reading them.